Advanced Calculus Single Variable

10.1 Finite p Variation

Instead of integrating with respect to a finite variation integrator function F, the function will
be of finite p variation. This is more general than finite variation.

Definition 10.1.1Define for a function F :

[0,T]

→ ℝ

α Holder continuous if

sup |F-(t)−-F-(s)| < C < ∞
0≤s<t≤T |t− s|α

Thus

|F (t) − F (s)| ≤ C|t− s|α

Finite p variation if for some p > 0,

( )1 ∕p
∑m p
∥F∥p,[0,T] ≡ suPp |F (ti+1)− F (ti)| < ∞
i=1

where P denotes a partition of

[0,T]

,P =

{t0,t1,⋅⋅⋅,tn}

for

0 = t0 < t1 < ⋅⋅⋅ < tn = T

also called a dissection.

|P|

denotes the largest length in any of the sub intervals. It willbe always assumed that actually p ≥ 1.

Note that when p = 1 having finite p variation is just the same as saying that it has finite
total variation. Thus this is including more general considerations. Also, to simplify the
notation, for P such a dissection, we will write

n
∑ |F (t )− F (t )|p instead of ∑ |F (t )− F (t)|p
P i+1 i i=1 i+1 i

Definition 10.1.2We use Cα

([0,T];ℝ)

to denote the α Holder functions andVp

([0,T],ℝ)

to denote the continuous functions F which have finite p variation.

It is routine to verify that if α > 1, then any Holder continuous function is a constant. It is
also easy to see that any 1∕p Holder is p finite variation. To see this, note that you
have